But it can be reduced to "The girl living next door is really pretty." Why can we reduced that to this but not to "The girl lives next door is really pretty", please?

Nouns may be postmodified by a number of items including relative clauses as well as the non-finite gerund-participial and past-participial clauses.

But "lives next door" is just a verb phrase, not a clause, and as such cannot serve as postmodifier of a noun.

Incidentally, I would urge you to avoid the term 'reduced relative clause'. In The girl [living next door] is really pretty, the bracketed element is not some kind of reduced relative clause, but a different kind of clause -- a non-finite gerund-participial one.

Tara2The train [just arrived at platform six] is the delayed 13.15 from Hereford.

Because "arrived" is a participle, this counts as a participle clause. It is correct, but it creates the same problem we've been talking about in this thread, namely that it tricks the reader into thinking the main verb comes earlier in the sentence than it actually does.

The train just arrived at platform six is a complete sentence. Then we see is the delayed 13.15 from Hereford, and we ask "What is that doing there?" So we have to go back to the beginning of the sentence and try to find another way to interpret it.

Some experts on grammar have given this kind of sentence a special name: a garden-path sentence. They "lead you down a garden path", an idiom which means to lead someone astray, to give someone false signals in order to deceive them.

In my opinion past participles that are identical to past tense forms cause too much confusion to be used after a noun like this. Good writers tend to avoid these patterns.

More examples:

The florist sent the flowers was pleased.The girl told the story cried. The raft floated down the river sank.